Enter Number(s)
Positive whole numbers only (2 or greater)
Examples:
What is Prime Factorization?
Prime factorization is the process of breaking any integer greater than 1 into its prime number building blocks. Every number has exactly one unique prime factorization — this is called the Fundamental Theorem of Arithmetic.
60 = 2 × 2 × 3 × 5 = 2² × 3¹ × 5¹
360 = 2³ × 3² × 5¹
1 = (special case — has no prime factors)
360 = 2³ × 3² × 5¹
1 = (special case — has no prime factors)
How to Find Prime Factorization (Trial Division)
- Start with the smallest prime: 2
- Divide the number by 2 as many times as possible
- Move to the next prime (3, 5, 7, 11...) and repeat
- Stop when the quotient becomes 1
- The primes you divided by are the prime factors
Divisor Count Formula
If n = p1^a × p2^b × p3^c
Number of divisors = (a+1) × (b+1) × (c+1)
Example: 60 = 2² × 3 × 5 → (2+1)(1+1)(1+1) = 12 divisors
Number of divisors = (a+1) × (b+1) × (c+1)
Example: 60 = 2² × 3 × 5 → (2+1)(1+1)(1+1) = 12 divisors
Uses of Prime Factorization
- Finding GCF: Multiply common prime factors with lowest exponents
- Finding LCM: Multiply all prime factors with highest exponents
- Simplifying fractions: Divide numerator and denominator by GCF
- Cryptography: RSA encryption relies on difficulty of factoring large numbers
- Number theory: Studying divisibility, perfect numbers, and more
Prime Factorization of Common Numbers
12 = 2² × 3 | 24 = 2³ × 3 | 36 = 2² × 3² | 48 = 2⁴ × 3
60 = 2² × 3 × 5 | 100 = 2² × 5² | 360 = 2³ × 3² × 5
Prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
60 = 2² × 3 × 5 | 100 = 2² × 5² | 360 = 2³ × 3² × 5
Prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Perfect Squares and Perfect Cubes
A perfect square has all prime factors with even exponents (e.g. 36=2²×3²). A perfect cube has all exponents divisible by 3 (e.g. 27=3³). This calculator detects both automatically and shows a badge.
How Prime Factorization Helps Find GCF and LCM
- GCF: Take common prime factors with the LOWEST exponent. GCF(12,18): 12=2²×3, 18=2×3² → GCF=2¹×3¹=6
- LCM: Take ALL prime factors with the HIGHEST exponent. LCM(12,18): LCM=2²×3²=36
- Verify: GCF×LCM=A×B → 6×36=216=12×18 ✓
Sum of Divisors Formula
If n = p1^a × p2^b then:
Sum of divisors = (1+p1+p1²+...+p1^a) × (1+p2+p2²+...+p2^b)
Example: 12=2²×3 → (1+2+4)×(1+3) = 7×4 = 28
Sum of divisors = (1+p1+p1²+...+p1^a) × (1+p2+p2²+...+p2^b)
Example: 12=2²×3 → (1+2+4)×(1+3) = 7×4 = 28
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